Integrate the rational function 2 x x^ 2 +3 x+2 - Vidyadip

Question

Integrate the rational function $\frac{2 x}{x^{2}+3 x+2}$

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Answer

Let $\frac{2 x}{x^{2}+3 x+2}=\frac{A}{(x+1)}+\frac{B}{(x+2)}$
$\Rightarrow$ 2x = A(x + 2)+ B(x + 1) …(i)
Substituting x = -1 and -2 respectively in equation (i), we get,
A = -2, B = 4
Thus,
$\frac{2 x}{x^{2}+3 x+2}=\frac{-2}{(x+1)}+\frac{4}{(x+2)}$
$\Rightarrow$ $\int \frac{2 x}{x^{2}+3 x+2} d x=\int\left\{\frac{-2}{(x+1)}+\frac{4}{(x+2)}\right\} d x$
= 4 log|x + 2| - 2 log|x + 1| + C

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